a model of broker's trading with applications to order flow internalization

8/7/2019 A Model of Broker's Trading With Applications to Order Flow Internalization

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A Model of Brokers Trading, With Applications to Order Flow

Internalization

Sugato Chakravarty

Purdue University

West Lafayette, Indiana, 47906

765-494-6427

Fax: 765-494-0869

EMAIL: [email protected]

Asani Sarkar

Federal Reserve Bank of New York

33 Liberty Street

New York, New York, 10045

212-720-8943Fax: 212-720-1582

EMAIL: [email protected]

Previous version: June 2000

Curr ent version: J anuar y 2001

The views expressed in this paper are those of the authors and do not represent those of the

Federal Reserve Bank of New York or the Federal Reserve System.


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A Model of Brokers Trading, With Applications to Order Flow

InternalizationJEL Classification number: G12

Abstract

Although brokers trading is endemic in securities markets, the form of this trading differs

between markets. Whereas in some securities markets, brokers may trade with their customers

in the same transaction (simultaneous dual trading or SDT), in other markets, brokers are only

allowed to trade aftertheir customers in a separate transaction (consecutive dual trading or

CDT). We show theoretically that, informed and noise traders are worse off and brokers are

better off while market depth is lower in the SDT market. Thus, given a choice, traders prefer

fewer brokers in the SDT market compared to the CDT market. With free entry, however,

market depth may be higher in the SDT market provided its entry cost is sufficiently low

relative to the CDT market. We study order flow internalization by broker-dealers, and show

that, in the free entry equilibrium, internalization hurts retail customers and market quality.


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Dual trading occurs when a broker sometimes trades for customers as an agent, and at

other times trades for his own account. Personal trading by brokers is pervasive throughout

securities and futures markets in the U.S. and the world. There are two types of dual trading:

simultaneous (where a broker trades for himself and a customer in the same transaction)and

consecutive (where a broker trades for customers as an agent and for himself at other times,

but not in the same transaction). 1

In futures exchanges, consecutive dual trading is permitted, but not simultaneous dual

trading. However, both types of dual trading are allowed in securities markets, currency and

interest rate swap markets, and the fixed income market. What are the relative advantages and

disadvantages of the two forms of dual trading, and how do they relate to the structure of the

securities markets? Our paper highlights the role of brokerage competition in determining the

relative costs and benefits of the two types of dual trading. We show that consecutive dual

trading markets are associated with many brokers, whereas simultaneous dual trading markets

are only viable with few brokers. Therefore, consistent with institutional reality, our model

predicts that in futures markets, where typically many dual traders are present in each contract,

consecutive dual trading is more likely than simultaneous dual trading. In stock markets, by

contrast, the number of dual traders in any stock is few, and simultaneous dual trading is the

norm.2

In our model, based on Kyle (1985), multiple informed traders and noise traders trade

by dividing their orders equally among multiple brokers. The brokers, on receiving the

1

The terminology is due to Grossman (1989), who also surveys dual trading regulations around the world.2

In the NYSE, for example, the potential dual traders are--in addition to the specialist--national full-line firms and


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orders, may dual trade consecutively or simultaneously. If brokers dual trade simultaneously,

each broker submits the net order (his personal trades plus his customers trades) to the market

maker for execution, and the game ends in a single period. If brokers trade consecutively,

trading occurs over two periods. In period one, each broker submits the sum of informed and

period-one noise trades to the market maker for execution. In period two, each broker submits

his personal trade and period two noise trades to the market maker. In all cases, the market

maker sets prices to make zero expected profits, conditional on observing the net order flow in

the market.

We find that the simultaneous dual trading markets are not viable when there are many

dual traders. The reason is that brokers mimic informed trades by trading with insiders in the

same direction, causing insiders to trade less and at a higher price (in absolute value), and

lowering informed profits.3 In addition, brokers offset noise trades, thereby increasing noise

trader losses. These effects are more adverse, the greater the number of brokers.

When both dual trading markets exist, informed traders have lower expected profits and

noise traders have higher losses with simultaneous dual trading. These results appear to imply

that, over time, simultaneous dual trading markets should cease to exist. To explain the

coexistence of the two types of dual trading, we extend the model by endogenizing the number

of brokers and informed traders in the market. In the free entry equilibrium, traders pay a

fixed fee to enter the market and, upon entering, choose the number of brokers to trade with.

We show that, provided the market entry fee is sufficiently low, enough informed traders enter

the investment banks. In 1989, there were six national full-line firms, and ten investment banks (Matthews, 1994).3

As a practical example of how brokers trading may hurt informed traders, large institutional customers in the

stock markets have long been concerned that brokers may use knowledge of their orders to trade for their ownaccounts. See Money Machine, Business Week, June 10, 1991, pages 81-84.


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the simultaneous dual trading market to make it viable. Further, traders choose many brokers

in the consecutive dual trading market, and only one broker in the simultaneous dual trading

market.

To test the applicability of our model, we study order flow internalization by brokers,

which primarily affects uninformed retail order flow (Easley, Kiefer and O' Hara (1996),

Battalio, Green and Jennings (1997), and the SEC (1997). We show that, in the free entry

equilibrium, internalization reduces market depth and price informativeness, and increases

uninformed losses. Further, the number of internalizing brokers is negatively related to the

market depth and the number of entering informed traders. Thus, our model predicts that

internalization may be more prevalent in thin markets with few informed traders. If thin

markets have high spreads, this result supports advocates of purchased order flow who argue

that it primarily affects NYSE stocks with large spreads (Easley, Kiefer and OHara, 1996).

However, contradicting these advocates, we show that market quality is affected adversely.

The existing literature on dual trading does not distinguish between the two types of

dual trading. Also, it does not consider multiple informed traders and multiple brokers in the

same model, nor does it endogenize the number of traders. Fishman and Longstaff (1992)

study consecutive dual trading in a model with a single broker and a fixed order size. Roell

(1990) and Chakravarty (1994) have multiple dual traders but a single informed trader. Sarkar

(1995) studies simultaneous dual trading with multiple informed traders and a single broker.

Regarding internalization, Battalio and Holden (1996) show that, if brokers can

distinguish between informed and uninformed orders (as in our model), they can profit from

internalizing uninformed orders. However, unlike our model, they do not focus on the effect

of internalization on market quality. Dutta and Madhavan (1997) find that a collusive


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equilibrium is easier to sustain with preferencing arrangements. In contrast to the theoretical

results, the empirical studies of Battalio (1997), Battalio, Greene and Jennings (1997) and

Lightfoot, Martin, Peterson, and Sirri (1997) and the experimental study of Bloomfield and

OHara (1996) find no adverse effect on market quality. 4

4

As Battalio, Greene and Jennings (1997) state, their result may imply that broker-dealers are not systematically

skimming uninformed order flows or, alternatively, that internalizing brokers have a cost advantage in executing

orders. Macey and OHara (1997) survey the literature on preferencing and internalization.

The remainder of the paper is organized as follows. Section I describes a trading model

with multiple customers and many brokers, when the number of informed traders and brokers

is fixed. Sections II and III solve the consecutive and simultaneous dual trading models.

Section IV endogenizes the number of informed traders and brokers. Section V analyzes order

flow internalization. Section VI concludes. All proofs are in the appendix.

I. A Model of Trading with Multiple Customers and Multiple Brokers.We consider an asset market structured along the lines of Kyle (1985). There is a

single risky asset with random value v, drawn from a normal distribution with mean 0 and

variance v. There are n informed traders who receive a signal about the true asset value and

submits market orders. For an informed trader i, i= 1,.. ,n, the signal is si = v + ei, where ei

is drawn from a normal distribution with mean 0 and variance e. A continuum of noise

traders also submit aggregate market orders u, where u is normally distributed with mean zero

and variance u. All random variables are independent of one another.


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All customers, informed and uninformed, must trade through brokers. There are m

brokers in the market, who submit customer orders to the market maker. m and n are common

knowledge. We assume that orders are split equally among the m brokers. By observing

informed orders, brokers can infer the informed traders' signals. By observing orders of noise

traders, they are aware of the size of uninformed trades. Consequently, brokers have an

incentive to trade based on their customers' orders. However, brokers are not allowed to trade

ahead of (i.e., front run) their customers.

Brokers may trade in two possible ways. They may execute their customers' orders

first, and trade for their own accounts second in a separate transaction---i.e., engage in

consecutive dual trading. Alternatively, they may trade with their customers in the same

transaction--i.e., engage in simultaneous dual trading. The sequence of events is as follows:

in stage one, informed trader i (for i= 1,..,n) observes si and chooses a trading quantity xi. In

stage two, broker j (for j= 1,.. ,m) trades consecutively or simultaneously, and places orders of

an amount zj

. In subsequent sections, we describe in more detail how simultaneous and

consecutive dual trading differ. Finally, all trades (including brokers' personal trades) are

batched and submitted to a market maker, who sets a price that earns him zero expected profits

conditional on the history of net order flows realized.

Initially, the number of informed traders and brokers is fixed. Later, we allow

informed traders to choose the number of brokers to allocate their orders to, and study the free

entry equilibrium where informed traders and brokers decide whether to enter the market,

depending on a market entry cost and their expected profits upon entering.


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II. Consecutive Dual Tr ading.

In this section, we solve for the equilibrium in a market with consecutive dual trading.

We assume that brokers do not trade with their customers in the same transaction--i.e.,

simultaneous dual trading is not allowed, as in futures markets.

A. The Consecutive Dual Trading Model

Trading occurs in two periods. In period one, brokers receive market orders from n

informed traders and the noise traders, which they then submit to the market maker. In period

two, brokers trade for themselves, along with period two noise traders. Each period, a market

marker observes the history of net order flow realized so far and sets a price to earn zero

expected profits, conditional on the order flow history.

The sequence of events is as follows: in period one, informed trader i, i= 1,..,n

observes si and chooses xi,d, knowing that his order will be executed in the first period.

Accordingly, informed trader i, i= 1,.. ,n, chooses xi,d to maximize conditional expected profits

E[(v-p1)xi,dsi], where the period one price is p1 = 1y1, the period one net order flow is y1 =

xd+ u1, the aggregate informed trade is xd = ixi,dand u1 is the period one noise trade.

In period two, brokers choose their personal trading quantity after observing the n-

vector of informed trades {x1,d, . . ,xn,d}, u1and p1. Thus, broker j, j= 1,.. ,m, chooses zjto

maximize conditional expected profits E[(v-p2)zj{x1,d/m ,..,xn,d/m}, u1/m, p1], where the

conditioning is based on each broker observing his portion of the informed and uninformed

orders received, plus the period one price.

The m brokers submit their personal trades to the market maker, who sets p2 =

2y2+ 2y1where y2 = jzj + u2, jzj is the aggregate trade of all brokers and u2is the noise


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trade in period two. Finally, the liquidation value v is publicly observed and both informed

traders as well as brokers realize their respective profits (if any).

Define t= v/s, where tis the unconditional precision ofsi, i= 1,.. ,n. Note that 0t1.

Further, define Q = 1+ t(n-1), where (Q-1)si represents informed trader i' s conjecture

(conditional on si) of the remaining (n-1) informed traders' signals. Since informed traders

have different information realizations, they also have different conjectures about the

information of other informed traders. talso measures the correlation between insider signals.

For example, ift= 1 (perfect information), informed signals are perfectly correlated, Q= n,

and informed trader i conjectures that other informed traders know (n-1)si--i.e., the informed

trader believes other informed traders have the same information as he.

Proposition 1 below solves for the unique linear equilibrium in this market.

Proposition 1: In the consecutive dual trading case, there is a unique linear equilibrium

for t> 0. In per iod one, informed trader i, i= 1,.. ,n, tr ades x i,d= Adsi, and the price is

p1= 1y1. In period two, each broker j, j= 1,.. ,m trades z = B1xd+ B2u1, the price is p2 =

2y2 + 2y1, and a brokers expected tr ading revenue is Wd where:

(1) +

u

v

Q

nt=

)1(1

(2) )1(1 Qt

=Ad+

(3) Q)+mQ(1=B1

1

(4)Q)+m(1

Q=B2


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(5)m+1

m

Q)+Q(1

nt=

u

v

2

(6)

u

v

2Q)+(1

nt=

(7)m)+(1m

1

Q)+Q(1

nt=W

vu

d

Discussion: Period one informed trades are not affected by dual trading in period two since

informed traders trade only in period one. The informed trading intensity Ad is positively

related to the market depth (1/1) and to the precision of the signal. In period two, dual traders

piggyback on period one informed trades (B1> 0) and offset noise trades (B2< 0). Competition

between informed traders leads insiders to use their information less, making piggybacking less

valuable. Higher values oftincreases the correlation between insiders' signals, reducing (for

n> 1) the value of observing multiple informed orders. Consequently, the extent of

piggybacking B1 decreases in the number of informed traders n and the information precision t.

Broker revenues increase with noise trading, and decrease with the number of brokers.

III. Simultaneous Dual Trading.

A. The Simultaneous Dual Trading Model

Simultaneous dual trading is modeled in a single period Kyle (1985) framework. The

notations are the same as in section II. All variables and parameters related to simultaneous

dual trading are denoted either with superscript s or subscript s.

A group ofn informed traders receive signals si about the unknown value v, and choose

quantities xi,s knowing that his order will be executed along with the orders of brokers and

noise traders in the same transaction. Accordingly, informed trader i, i= 1,.. ,n, chooses xi,s to


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maximize conditional expected profits E[(v-ps)xi,ssi], where the price is ps = sys, the net order

flow is ys = xs+ mz+ u, the aggregate informed trade is xs= ixi,s , z is the amount each broker

trades and u is the noise trade.

Upon receiving the orders of their informed and uninformed customers, brokers choose

their personal trading quantity after observing the n-vector of informed trades {x1,s, . . ,xn,s} and

u. Thus, broker j, j= 1,.. ,m, chooses zj,s to maximize expected profits E[(v-

ps)zj,s{x1,s/m,..,xn,s/m}, u/m], where the conditioning is based on each broker observing his

portion of the informed and uninformed orders.

The m brokers submit their customer trades and personal trades to the market maker,

who sets the price that earns him zero expected profits conditional on the net order flow

realized. Finally, the liquidation value v is publicly observed and both informed traders as

well as brokers realize their respective profits (if any). Proposition 2 below solves for the

unique linear equilibrium in this market.

Proposition 2: In the simultaneous dual trading case, there is a unique linear equilibrium

for t> 0 and Q> m. Informed trader i, i= 1,.. ,n, tr ades x i,s= Assi, br oker j, j= 1,.. ,m

trades z = B1,sxs+ B2,su, the pr ice is ps = sys, Ws is the brokers expected revenue, where:

(8)Q)+(1

ntm)+(1=

u

v

s

(9))1(

)(

QQ

tmQ=A

s

s +

(10)m)-(Q

1=B s1,

(11)m)+(1

1-=B s2,

(12)m)+Q(1

nt=W

vus


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Discussion. In contrast to Proposition 1, the informed trading intensity As depends on the

number of brokers. The insider trades and equilibrium exists only ifQ> m. Since n> Q,

existence implies n> m: the number of informed traders must exceed the number of brokers.

The intuition behind this result is as follows. Suppose an insider buys. Dual traders also buy

in the same transaction, piggybacking on the insider trade (i.e. B1> 0), and increasing the price

paid by the insider for his purchase. The order of an individual insider is exploited less as the

number of insiders increases, and is exploited more as the number of brokers increases. If the

number of brokers is too large relative to the number of insiders, the adverse price effect

makes it too costly for insiders to trade.

B. Comparing Simultaneous And Consecutive Dual Trading

By combining results from propositions one and two, we can compare the equilibrium

outcomes for the two kinds of dual trading, holding the number of informed traders and

brokers fixed.

Corollary 1. Suppose m and n is fixed. Then:

(1) If Q =m, only the consecutive dual trading equilibrium is viable.(2) If Q> m, then both dual trading equilibria are viable. Relative to simultaneous dual

trading, uninformed losses and brokers profits are lower, while informed profits are higher

with consecutive dual trading.

Discussion. As discussed in Proposition 2, the simultaneous dual trading equilibrium no

longer exists when Qm, i.e. when there are too many brokers who take advantage of the

insider information. Since, with consecutive dual trading, informed trading is independent of

the number of brokers, a market with consecutive dual trading continues to exist even with too


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many brokers. When the number of brokers is relatively few (Q> m), then both types of dual

trading exists. Brokers would prefer simultaneous since their profits are higher. Since

brokers profits are at the expense of insiders, informed traders by contrast prefer consecutive

dual trading. Aggregate profits of informed traders and dual traders are higher in the

simultaneous dual trading market. Since noise trader losses are the negative of these aggregate

profits, noise trader losses are also higher with simultaneous dual trading.

IV. Free Entry by Informed Traders and Brokers

In this section, we study the two forms of dual trading, given that informed and

uninformed traders optimally choose the number of brokers to give their orders to, and given

that there is free entry into the asset market. The decision-making sequence of agents is as

follows: 1) Informed traders and brokers simultaneously decide whether to enter the market;

(2) Informed and noise traders choose the number of brokers to give the order to; (3) Informed

and noise traders divide their orders equally among the chosen brokers. From then on, the

game continues as before.

Informed traders choose the number of brokers to maximize expected profits.

Uninformed noise traders choose the number of brokers to minimize their expected losses to

informed traders and brokers. The following corollary describes traders choice of the number

of brokers to trade with.

Corollary 2. (1) When brokers tr ade consecutively, informed and noise traders choose all

the brokers available.


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(2) Suppose there are at least two informed traders, so that the simultaneous dual trading

equilibr ium exists. Then, with simultaneous dual trading, informed and noise tr aders

give orders to only one broker.

Discussion. With consecutive dual trading, informed traders' profits are independent ofm but

noise trader losses are decreasing in m. Thus, noise traders choose all available brokers while

informed traders are indifferent to the choice ofm. When brokers trade simultaneously, the

situation is reversed: informed profits are decreasing in m whereas uninformed losses are

independent ofm. Thus, informed traders choose one broker and noise traders are indifferent

to the choice ofm. For concreteness, we assume that informed and noise traders give orders

to the same number of brokers.

Next, consider free entry by informed traders and brokers. Let ki be the cost of

entering market i, i= d(consecutive dual trading), s (simultaneous dual trading). To obtain

analytic solutions, we assume t= 1. The free entry equilibrium satisfies two conditions.

Traders enter a market until their expected profits, net of the entry cost, are zero. And the

cost is low enough so that entry is profitable for the minimum number of traders necessary to

sustain equilibrium. Proposition three solves for the free entry equilibrium in the two markets.

Pr oposition 3. (1) In the market with consecutive dual tr ading, nd informed traders and

mdbrokers enter the market, with md< nd, and md and nd given by:

(13) n+1

1

k=)m+(1m dd

vu

dd

(14)k

=)n+(1nd

vu

dd

At least two informed traders and at least one broker enter the market if:


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(15)23

kvu

d

(2) In the market with simultaneous dual trading, ns informed traders and one broker

enter the market, where ns is given by:

(16)k2

=1)-n(

)n+(1nn

s

vu

s

sss

At least two informed traders and one broker enter the market if:

(17)212

kvu

s

Discussion. Proposition three is intuitive: the number of informed traders and brokers

entering a market is inversely related to the cost of entering the market, and positively related

to the volatility of the asset value and the size of noise trades (i.e. , the depth of the market). In

the consecutive dual trading market, a broker expects to make less trading profits than an

informed trader since he trades second. Thus, the equilibrium number of brokers is less than

the equilibrium number of informed traders in this market.

An important observation is that the participation constraint (17) is more restrictive than

the inequality (15), implying that the simultaneous dual trading equilibrium is viable only at

lower market entry cost, relative to the consecutive dual trading equilibrium. In the following

proposition, we compare the two types of dual trading markets given free entry by brokers and

informed traders and the optimal choice of the number of brokers by informed traders.

Pr oposition 4. In the free entr y equilibr ium of proposition three, suppose kd > 2ks. Then

nd < ns. Market depth may be higher with simultaneous dual trading ifns is large

relative to nd and md.


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Discussion. The proposition says that, if entry costs are sufficiently low, the simultaneous

dual trading market may have more informed traders and greater market depth in a free entry

equilibrium. The reason is that informed traders protect themselves from excessive

piggybacking by choosing only one broker, thus maximizing their expected profits in the

simultaneous dual trading market. Consequently, if entry costs are low enough, many

informed traders enter the simultaneous dual trading market and market depth is higher. Thus,

the proposition provides some intuition as to why we see markets with simultaneous dual

trading exist in the real world.

V. Internalization of order flow by br oker-dealers

Internalization is the direction of order flow by a broker-dealer to an affiliated specialist

or order flow executed by that broker-dealer as market maker. Broker-dealers can internalize

order flow in several ways. For example, large broker-dealer firms, particularly NYSE

member firms, purchase specialist units on regional exchanges and direct small retail customer

orders to them. In off-exchange internalizations, NYSE firms execute orders of their retail

customers against their own account, with the transaction taking place in the so-called third

market or over-the-counter market. Such transactions, also called 19c-3 trading, have become

a major source of profits for broker-dealers. 5

We use our simultaneous dual trading model to analyze order flow internalization.

Since internalizing brokers typically handle order flows of small retail investors, we assume

that there are m1 internalizing brokers who handle all of the noise trades, and m2 piggybacking

5

Rule 19c-3 allows NYSE stocks listed after April 26, 1979 to be traded off-exchange. Broker-dealers may have

earned over $500 million in 1994 from 19c-3 trading. See Report on the practice of preferencing, the SEC,April 1997, and In-House trades can be costly for small investors, the Wall Street Journal, December 20, 1994.


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brokers who handle all the informed traders, with m1+ m2 = m. As before, the market maker

sees the pooled order and, further, cannot distinguish between internalizing brokers and

informed-order brokers.

Let z1be the trade of an internalizing broker and let z2 be the trade of informed-order

brokers. From (10) and (11):

(18))m+(1

u-=z

1

1

(19))m-(Q

x=z

2

s2

We compare uninformed losses and the market quality between the internalization

model and a model with no brokers' trading. For simplicity, we assume t= 1. The following

proposition shows the effect of order flow internalization on noise trader losses and the market,

assuming free entry of informed traders and brokers.

Pr oposition 5. Suppose mar ket entr y is free. Then, with internalization of order flow, (i)

one piggybacking broker, noinformed traders and mointernalizing brokers enter the

market. mo is less than the competitive number of brokers since internalizing brokers

make positive expected pr ofits in the free-entr y equilibrium. (ii) Relative to a mar ket with

no order flow internalization, market depth, price informativeness, expected informed

profits and the number of informed traders are lower, while uninformed losses are

higher.

Discussion. Since piggybacking hurts informed profits, traders choose just one piggybacking

broker. Aggregate profits of internalizing brokers are greater than the aggregate profits of

informed traders. Since entry costs must be low enough to allow informed traders to enter the


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market at zero expected profits, it follows that expected profits of internalizing brokers must be

positive at these costs. The results on market quality and informed profits follow from our

earlier result that market quality in the simultaneous dual trading market is worse than in the

consecutive dual trading market and in a market without dual trading.

VI. Conclusion.

In this article, we study a wide variety of issues related to brokers trading. Multiple

informed traders and noise traders trade through multiple brokers, who either trade in the same

transaction as their customers (simultaneous dual trading) or in a separate transaction

(consecutive dual trading).

While the consecutive dual trading equilibrium always exists, the simultaneous dual

trading equilibrium fails when the number of brokers is greater than the number of informed

traders. The reason is that brokers trade with informed traders in the same direction, thus

worsening informed traders terms of trade. This effect is magnified with many brokers,

leading informed traders to stop trading. When both equilibria exist, informed profits and

market depth are lower, while uninformed losses and brokers profits are higher with

simultaneous dual trading, relative to consecutive dual trading. Thus, only brokers prefer

simultaneous dual trading.

We allow informed and noise traders to choose the number of brokers, and endogenize

the number of brokers and informed traders in the market. In the simultaneous dual trading

market, informed and noise traders choose only one broker whereas, with consecutive dual

trading, informed and noise traders choose all available brokers. If the market entry cost is

sufficiently low, more informed traders enter the simultaneous dual trading market, and market


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depth may be lower, relative to the consecutive dual trading market. Thus, the adverse effects

of simultaneous dual trading on customers and the market are mitigated in the free entry

equilibrium. If the market entry cost is high, consecutive dual trading is better.

In the simultaneous dual trading model, we allow some brokers to internalize the

uninformed order flow, by selling to noise traders as dealers, out of inventory. In the free

entry equilibrium, we find that market depth and price informativeness are lower, and

uninformed losses are higher. In addition, the number of internalizing brokers is negatively

related to market depth and the number of informed traders.


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Battalio, R., Greene, J. and R. Jennings (1997), Do competing specialists and preferencing

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Appendix

Proof of Proposition 1:

Consecutive dual trading equilibrium.

In period one, the insiders problem is identical to the single period model in Kyle (1985), but

with multiple informed traders. The solution to this equilibrium is in Lemma 1 of Admati and

Pfleiderer (1988). This gives Ad and 1.

The j-th dual traders problem:

Let xm,d denote the n-tuple {x1,d/m, x2,d/m,.., xn,d/m}. In period two, dual trader j observes

xm,d, p1 and u m1/ and trades z j . The dual traders problem is to maximize with respect to zj

his expected profits E[(v-p2)zjxm,d, u1/m, p1]. Let jz = izi for all ij. Also, define=

=n

i

iss

1

.

The first order condition for the dual traders problem is:

(1.1) 02 12222, = usAzzxvE d

jjdm

Using linear projection, E[vxm,d

] = ts/Q, where Q is defined in the text. The second order

condition is -1 < 0, which is satisfied for 1 > 0.

Given that brokers are symmetric, z = zj for every j. Then, solving for z from (1.1),

(1.2) 12

2

21 )1()1()1(u

mm

T

QQ

tsz

+

++=

where

(1.3) QQT 21 )1( +=

Now, y2 = jzj + u2 = mz + u2. Substituting for z from (1.2), we have:

(1.4) 2121

2)1(

uuTTTQQ

tsy +

+=

where


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(1.5))1(2 m

mT

+=

Since the period one net order flow y1= Ads + u1, we can substitute for Ad from the text and

rewrite as:

(1.6) 11

1)1(

uQ

tsy +

+=

Let 01= Cov(v,y1), 02= Cov(v,y2), 12= Cov(y1,y2), 11= Var(y1), and 22= Var(y2). From

linear projection, 1, 2 and 2 are given by the following formulas:

(1.7)11

01

1

=

(1.8)D

12011102

2

=

(1.9)D

12022201

2

=

(1.10) ( )2122211 =D

The results of the Proposition can be obtained after working out the expressions for ij,

i,j= 1,2 and substituting in (1.7)-(1.10). A less computation-intensive alternative is to assume

12= 0, which implies (from 1.7 and 1.9) 2= 01/11= 1. From (1.8), 2= 02/22 which gives

the expression in the text. To complete the proof, we check that if2 and 2 are as given in

the proposition, 12= 0.

Having obtained 2 and 2, we substitute in (1.2). In the resulting expression, the coefficient

of xd is B1 and the coefficient of u1 is B2. This gives (3) and (4) in the text.

Finally, the expected profits of the j-th broker is Wd= E[(v-p2)z]. After substituting for p2:

(1.11) [ ]zuxumzvEW dd ))()(( 1222 ++=


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22

After appropriate simplifications, we get (7) in the text.


Simultaneous dual trading equilibrium.

Denote xs ==

n

i

six

1

, . There is only one period in this case, with net order flow:

(2.1) uxzy s

m

j

j

s ++==1

Let xm,s denote the n-tuple {x1,s/m, x2,s/m,.., xn,s/m}. Dual trader j observes xm,s, and u and

trades z j . The dual traders problem is to maximize with respect to zj his expected profits

E[(v-ps)zjxm,s, u/m]. The first order condition for the dual traders problem is:

(2.2) 02, = uxzzxvE sssj

s

j

s

ds

Using linear projection, E[vxs,d] = (txs/QAs). The second order condition is -s < 0, which is

satisfied for s > 0.

Given that brokers are symmetric, z = zj for every j. Then, solving for z from (2.2),

(2.3))1()1( m

u

QA

t

m

xz s

ss

s

+

+=

Informed trader iobserves si and trades xi,s. The informed traders problem is to maximize

with respect to xi,s his expected profits E[(v-p) xi,ssi]. Let x-i,s = k xk,s for all ki. Also, note

from linear projection that E[vsi] = tsi and and E(x-i,s|si] = (n-1)Astsi. The first order condition

for the informed traders problem is:

(2.4) [ ] 0)1()1(11

2 , =++

+

++

+ QnAmQ

mQ

ts

QA

t

m

m

mx ss

i

s

ssi

The second order condition is s + (mt/QAs) > 0. Given s> 0, this requires As> 0.


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We solve for xi,s from (2.4). In the resulting expression, As is the coefficient of si and this

gives us (9) in the proposition.

We substitute for As in (2.3). In the resulting expression, B1,s is the coefficient of xs and B2,sis

the coefficient of u. This gives (10) and (11) in the proposition. s is obtained from the

formula s = Cov(v,ys)/ Var(ys). Finally, the expected profits of the j-th broker is obtained

from the expression Ws= E[(v-ps)z].

Proof of Corollary 1:

(1)When Qm, then As0 and so the simultaneous dual trading equilibrium does not exist.However, the consecutive dual trading equilibrium is viable so long as m1.

(2)Suppose Q> m. In the consecutive dual trading equilibrium, profits for informed trader iare Ii,d = E[(v-p1)xi,d]. Aggregate informed profits are Id= nIi,d. Using the results from

Proposition 1 and substituting:

(c1.1) )1( Q

nt

I

vu

d+

=

In the simultaneous dual trading equilibrium, profits for insider i are Ii,s= E[(v-ps)xi,s].

Aggregate informed profits are Is= nIi.s. Using the results from Proposition 2 and substituting:

(c1.2)m

mQ

QQ

ntI

vu

s +

+

=

1)1(

Since (Q-m)/Q < 1 and 1/(1+ m) < 1, Is

< Id

.

Noise trader losses in the consecutive dual trading equilibrium Ud is the sum of aggregate

informed and dual trading profits (since the market maker makes zero expected profits by

assumption). Thus Ud= Id+ mWd. Substituting for Id from (c1.1) and for Wd from the text,


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(c1.3)

++

++

=

m

m

Q

Q

Q

ntU

vu

d1

11

1

Noise trader losses in the simultaneous dual trading equilibrium Us= Is+ mWs. Substituting for

Is from (c1.2) and for Ws from the text,

(c1.4)Q

ntU

vu

s +

=

1

Clearly, Ud > Us. The result for broker profits follows from direct comparison of (7) and (12)

in the text.

Proof of Corollary 2:

(1)From (c1.1) in Corollary 1, Id is independent of m. From (c1.3) in Corollary 1, Ud isdecreasing in m. Thus noise traders choose the highest possible number of brokers and we

assume informed traders do the same.

(2)From (c1.2) in Corollary 1, Is is decreasing in m and, from (c1.4) in Corollary 1, Us isindependent of m. Thus, informed traders choose m= 1 and we assume noise traders do

the same.


1) Since we assume t= 1, Q= n. Since Id is decreasing in n, the equilibrium number of

informed traders nd satisfies Id= nkd. Using (c1.1) in Corollary 1, we have:

(3.1)d

vu

ddk

nn

=+ )1(

Similarly, the equilibrium number of brokers md satisfies Wd= kd and, using (7) in the text, we

have:


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(3.2)dd

vu

dd

nkmm

+

=+

1

1)1(

From (3.1) and (3.2), md < nd.

If md= 1 and nd = 2 is to be an equilibrium, then Idnkd and Wdkd at these values, so that from

(3.1) and (3.2) we have:

(3.3)23

vu

dk

(3.4)32

vu

dk

Clearly, if (3.3) is satisfied, so is (3.4).

2) In the simultaneous dual trading equilibrium, ms= 1. Given ms= 1, the condition Is= nks

implies, from (c1.2) in Corollary 1, that:

(3.5)s

vu

s

sss

kn

nnn =

+

1

)1(2

If ms= 1 and ns = 2 is to be an equilibrium, then Isnks and Wsks at these values, so that from

(3.5) above and (12) in the text:

(3.6)212

vu

sk

(3.7)22

vu

sk

Clearly, if (3.6) is satisfied, so is (3.7).


Assume (3.6) above in Proposition 3 is satisfied. Then, with free entry, both consecutive and


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simultaneous dual trading equilibria exist. From (3.1) and (3.5) in Proposition 3:

(4.1)s

d

s

s

d

s

d

s

k

k

n

n

n

n

n

n

2

1

1

1 =

++

If kskd/2, then (4.1) implies that nsnd.

From (5) and (8) in the text,

(4.2)s

d

s

s

d

d

d

s

n

n

n

n

m

mm

+

+

+

++=

1

1

1

112

2

If md and ns are both large, then (1+ md)/ md and ns/(1+ ns) are approximately 1, and (4.2) is

approximately:

(4.3)s

d

d

s

n

nm

+

++

1

112

2

From (4.3), s< 2 if ns is large relative to nd and md.


(i) We use the subscript o to refer to outcomes in the order flow internalization model. Let Ao

be the informed trading intensity, o be the (inverse of) market depth, Io be aggregate expected

informed profits, m1Wo be the aggregate expected profits of the internalizing brokers and Uo be

uninformed losses. Analogous to Proposition 2, we can show that:

(5.1)v

u

on

t

mQ

mQA

+

=)1( 1

2

(5.2)u

v

oQ

ntm

+

+=

)1(

11

(5.3))1)(1( 1

2

mQ

nt

Q

mQI

vu

o ++

=


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(5.4))1)(1( 1

11mQ

ntmWm

vu

o ++

=

(5.5))1( Q

ntU

vu

o

+

=

We know from Proposition 3 that the number of piggybacking brokers m 2= 1 in equilibrium.

Further, equilibrium requires n> m2, so that n2. The number of internalizing brokers m1 is

determined in equilibrium. Let mo be the equilibrium number of internalizing brokers. Let ko

be the entry cost in the market. Finally, let t= 1 so that Q= n. Thus, the equilibrium number of

informed traders no satisfies Io= nko and (5.3) implies:

(5.6)oo

o

oo

vu

omn

n

nnk

++

=1

1

1

1

Given no and m2= 1, mo satisfies Woko and (5.4) implies:

(5.7)oo

o

vuomn

nk

++

1

1

1

Inspection of (5.6) and (5.7) shows that both expressions cannot hold as equalities. So (5.7)

must hold as an inequality, implying Wo> ko In other words, the brokerage market is not

competitive, since brokers make positive expected profits even with free entry.

(ii) In the market without internalization, and with free entry, m2= 1 and m1= 0. The

equilibrium number of informed traders nw satisfies Iw= nko where Iw is Io evaluated at m2= 1

and m1= 0.

(5.8)1

1

+

=w

w

ww

vu

on

n

nnk

Comparing (5.6) and (5.8), no< nw.


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Let w be the (inverse of) market depth in the market without internalization. w is simply o

evaluated at m1= 0. Using (5.2), we have:

(5.8) 1

1

)1(

)1(

12

+

+

+

= ww

oo

ww

o

o

n

n

nn

nnn

o > w since nw > no 2.

Price informativeness with internalization, PIo, is defined as v-Var(vpo), where po = oyo

and yo is the net order flow with internalization. It can be shown that PI = nv/(1+ n).

Therefore, nw > no implies that PIo = nov/(1+ no) < nwv/(1+ nw) = PIw (price

informativeness without internalization).

Expected informed profits without internalization Iw is Io evaluated at m1= 0. From (5.3),

Io< Iw.

From (5.5), uninformed losses with internalization are proportional too

o

n

n

+1whereas

uninformed losses Uw without internalization are proportional tow

w

n

n

+1 . Hence, Uo> Uw.

a model of broker's trading with applications to order flow internalization

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